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  • 1
    Electronic Resource
    Electronic Resource
    On the classical Tauberian theorems (1963)
    Springer
    Archiv der Mathematik 14 (1963), S. 317-322 
    Details
    ISSN: 1420-8938
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01234960
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    On the Berry-Esseen theorem (1968)
    Springer
    Probability theory and related fields 10 (1968), S. 261-268 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The paper presents a simple derivation of a generalized Berry-Esseen theorem not requiring moments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00536279
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  • 3
    Electronic Resource
    Electronic Resource
    Limit theorems for probabilities of large deviations (1969)
    Springer
    Probability theory and related fields 14 (1969), S. 1-20 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Let {X k,n ; k=1, ⋯, n} be a triangular array of independent variables with row sums S n . Suppose E(X k,n ) = 0 and E(S n 2 ) = 1 and that $$\psi _n (h) = \sum\limits_{k = 1}^n {\log E(e^{h X_{k,n} } )}$$ exists for 0≦h≦ɛ n . Under mild conditions we show that (1) $$P\{ S_n 〉 z_n \} = \exp [ - r_n + 0(r_n )], r_n \to \infty$$ where the quantities z n and r n are related by the parametric equations (2) $$z_n = \psi '_n (h_n ), r_n = h_n \psi '_n (h_n ) - \psi _n (h_n ).$$ If the distributions of the X k,n behave reasonably well it is usually not difficult to obtain satisfactory asymptotic estimates for z n in terms of r n and vice versa. The principal application is to sequences X k . Then X k,n = X k /s n and S n = (X 1+⋯.+X n )/s n . A familiar special case of (1) is given by $$P\{ X_1 + \cdots + X_n 〉 s_n z_n \} \sim [1 - \mathfrak{N}(z_n )] \exp [ - P_n (z_n )]$$ where $$\mathfrak{N}$$ is the standard normal distribution and P n a certain power series. In this case rn = z n 2 but (2) may lead to radically different relationships between rn and zn.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00534113
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    General analogues to the law of the iterated logarithm (1969)
    Springer
    Probability theory and related fields 14 (1969), S. 21-26 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Let X1, X2,⋯. be independent and Sn=X1+⋯.+Xn. Suppose E(Xk)=0 and s n 2 = E(S n 2 )〈t8 and put $$\psi _\mathfrak{n} (h) = \log E(e^{hS_n /\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\cdot}$}}{s} _n } )$$ for h〉0. To avoid trivialities we suppose that Sn+1〈sn(log sn)p for some p〉0. We assume that there exists a number q〈1 such that if $$q\log \log s_n 〈 \zeta _n 〈 q^{ - 1} \log \log s_n$$ there exists a (necessarily unique) number x n (ζ n ) determined by the parametric equations $$\zeta _n = h\psi '_n (h) - \psi _n (h), x_n \zeta _n = \psi '_n (h).$$ It was shown in the preceding paper that under mild restrictions on the X n $$P\{ S_n 〉 s_n x_n (\zeta _n )\} = \exp [ - \log \log s_n + 0(\log \log s_n )].$$ It is now shown that under these conditions with probability one $$\lim \sup \frac{{S_n }}{{s_n x_n (\zeta _n )}} = 1.$$ It is easy to give examples where $$x_n \zeta _n \sim C(\log \log x_n )^{\tfrac{1}{2}}$$ with an arbitrary C〉0. Other examples, however, exhibit an entirely different behavior of the sequence }s n x n (ζ n )}.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00534114
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  • 5
    Electronic Resource
    Electronic Resource
    Book reviews (1956)
    Springer
    Psychometrika 21 (1956), S. 217-218 
    Details
    ISSN: 1860-0980
    Source: Springer Online Journal Archives 1860-2000
    Topics: Psychology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02289101
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  • 6
    Book
    Book
    ¬An¬ introduction to probability theory and its applications (1971)
    New York u.a. :Wiley,
    Details
    Title: ¬An¬ introduction to probability theory and its applications
    Author: Feller, William
    Publisher: New York u.a. :Wiley,
    Year of publication: 1971
    Pages: 669 S.
    Series Statement: Wiley-Series in Probability and Mathematical Statistics
    Type of Medium: Book
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    Zuse Institute Berlin get availability status
  • 7
    Book
    Book
    ¬An¬ introduction to probability theory and its applications (1968)
    New York u.a. :Wiley,
    Details
    Title: ¬An¬ introduction to probability theory and its applications
    Author: Feller, William
    Publisher: New York u.a. :Wiley,
    Year of publication: 1968
    Pages: 509 S.
    Series Statement: Wiley-Series in Probability and Mathematical Statistics
    Type of Medium: Book
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    ZIB Catalog
    Zuse Institute Berlin get availability status
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